SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

It is clear that the SV has a constant magnitude of 3 k /2 and its instantaneous angular position is
ωt. As time increases it will trace a circular trajectory in time. Its projection (given by Equation
3.72) onto the αβ plane is Equation (3.94).
3
x k X p cos( t)
2
3
x k X p sin( t)
2
(3.94)
The phase variables, stationary variables, and SV trajectory in the αβ plane are all shown in
Figure 3.30 for the case of balanced sinusoidal phase variables. The instantaneous position of the
quantities is shown at t 60 .
Figure 3.30 – Waveforms and SV trajectory for balanced sinusoidal phase variables.
It is clear that the magnitude of the SV (thus the amplitude of the α and β projections) is k 3/2
larger than the amplitude of the sinusoidal phase variables. The factor of 3/2 is the same one
that has been encountered before. The scaling coefficient k is artificially added to correct for the
differences between the three-phase and two-phase models. k is rarely left explicit in formulae,
rather a value is chosen to suit a particular purpose. The most obvious choice is k 1.
This
choice has appeal in dealing with torque and MMF because it yields SV equations that are similar
to the traditional expressions (as seen throughout Part II). Choosing k 2/3 will yield a SV
magnitude that is equal to the amplitude of the phase variables. Substituting this value into the
Clarke and inverse Clarke matrices thus yields a magnitude invariant transformation. A third
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